Face numbers of uniform triangulations of simplicial complexes

A triangulation of a simplicial complex $\Delta$ is called uniform if the $f$-vector of its restriction to a face of $\Delta$ depends only on the dimension of that face. This paper proves that the entries of the $h$-vector of a uniform triangulation of $\Delta$ can be expressed as nonnegative integer linear combinations of those of the $h$-vector of $\Delta$, where the coefficients depend only on the dimension of $\Delta$ and the $f$-vectors of the restrictions of the triangulation to simplices of various dimensions. Moreover, it provides information about these coefficients, including formulas, recurrence relations and various interpretations, and gives a criterion for the $h$-polynomial of a uniform triangulation to be real-rooted. These results unify and generalize several results in the literature about special types of triangulations, such as barycentric, edgewise and interval subdivisions.